An intermediate phase induced by dilution in a correlated Dirac Fermi system Lingyu Tian1 2Jingyao Meng1 2and Tianxing Ma1 1Department of Physics Beijing Normal University Beijing 100875 China

2025-04-27 0 0 917.08KB 6 页 10玖币

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An intermediate phase induced by dilution in a correlated Dirac Fermi system

Lingyu Tian,1, 2 Jingyao Meng,1, 2 and Tianxing Ma1, ∗

1Department of Physics, Beijing Normal University, Beijing 100875, China

2Beijing Computational Science Research Center, Beijing 100193, China

Substituting magnetic ions with nonmagnetic ions is a new way to study dilution. Using determinant quantum

Monte Carlo calculations, we investigate an interacting Dirac fermion model with the on-site Coulomb repulsion

being randomly zero on a fraction xof sites. Based on conductivity, density of states and antiferromagnetic

structure factor, our results reveal a novel intermediate insulating phase induced by the competition between

dilution and repulsion. With increasing doping level of nonmagnetic ions, this nonmagnetic intermediate phase

is found to emerge from the zero-temperature quantum critical point separating a metallic and a Mott insulating

phase, whose robustness is proven over a wide range of interactions. Under the premise of strongly correlated

materials, we suggest that doping nonmagnetic ions can effectively convert the system back to the paramagnetic

metallic phase. This result not only agrees with experiments on the effect of dilution on magnetic order but

also provides a possible direction for studies focusing on the metal-insulator transition in honeycomb lattice like

materials.

I. INTRODUCTION

Since a number of exotic phenomena are observed in

graphene[1–3] and silicene[4–6], the honeycomb lattice has

become another topic that has received much theoretical[7,8]

and experimental[9,10] attention with respect to the square

lattice. In the absence of interactions, itinerant electrons on

a honeycomb lattice form a Dirac spectrum, and the den-

sity of state (DOS) near Fermi energy, E=0, vanishes lin-

early, which is in strong contrast with the square lattice whose

DOS diverges at E=0[11]. The difference is directly reﬂected

in the onset of long-range antiferromagnetic (AFM) corre-

lations at half ﬁlling: the AFM order can be exhibited in

the square lattice for arbitrarily small on-site Coulomb re-

pulsion U, whereas a ﬁnite value Uc∼4tis required in the

honeycomb lattice[12,13]. In addition, a transition from a

Dirac semimetal phase to an insulating phase is also found

at Uc[14,15]. The massless Dirac fermions have advanced

our understanding of physics beyond Landau’s theory of the

Fermi liquid[16,17], which states that interacting metallic

systems are similar to free Fermi systems.

In real materials, disorder is inevitably present and can be

controlled by doping. Disorder plays an important role in

many novel physical properties of modern science, touch-

ing upon topics from transport phase transition[18–20] and

superconductivity[21,22] to quantum spin liquids[23–26].

However, different types of disorder may have opposite in-

ﬂuences on the physical mechanism under the same model,

which makes studies about disorder in correlated systems im-

portant and interesting. For example, the nearest-neighbor

hopping disorder is proven to enhance localization in the

two-dimensional repulsion Hubbard model under half ﬁll-

ing, while the site disorder reduces this effect. Interestingly,

both types of disorder destroy the AFM order for dynamic

properties[27]. Another example is using strong-coupling per-

turbation theory to study the Anderson Hubbard model on

the honeycomb lattice, where an intermediate metallic state

∗txma@bnu.edu.cn

is present between the Anderson insulator and Mott insulator

under binary-alloy disorder but absent under uniformly dis-

tributed disorder[28,29].

Site dilution is the disorder achieved by substitution of

magnetic ions with nonmagnetic ions. It has been reported

that doping nonmagnetic Pt and G into GaFe4As3produces a

different modiﬁcation of the electronic structure and transition

property[30]. In an insulating honeycomb magnet α−RuCl3,

replacing the magnetic ions Ru3+with nonmagnetic ions Ir3+

suppresses the magnetic order and induces a dilute quantum

spin liquid state in the low-temperature region [31–33]. Many

models have been used to reveal the phases in diluted systems

[34],and, motivated by cuprates such as La2CuxMg1−xO4, the

site diluted Hubbard model with on-site repulsion Ubeing

zero randomly on a fraction xof sites has been investigated in

two and quasi-two dimensions. On the strong coupling square

lattice, the AFM magnetic order at half ﬁlling disappears at

xc, which is consistent with the classical percolation thresh-

old x(perc,square)

c, while on the Lieb lattice, xcis almost twice

that of x(perc,lieb)

c[35]. This difference emphasizes the central

role of electron itinerancy in the magnetic response. In related

experiments, the honeycomb lattice provides a great platform

to study dilution. In CoTiO3, a linear relation between di-

lution and the critical temperature of magnetic transition is

observed over a wide dilution range[36]. Under out-of-plane

interactions or second- and third-nearest-neighbor exchange

interactions, the long-range AFM order survives well past the

classical percolation threshold xc=0.3[36–38]. Our study fo-

cuses on the magnetic phase transition caused by dilution in

the system with on-site interaction, and it is also interesting to

study whether magnetic order transition and metal-insulator

transition take place concomitantly.

Here, we use determinant quantum Monte Carlo (DQMC)

simulations to examine the effect of dilution on the ground

state properties of honeycomb lattices, including transport and

magnetic properties, of the half-ﬁlled case. Our key result

is that an intermediate gapped insulating phase without mag-

netic order is identiﬁed between the AFM Mott insulator and

the metal and is robust over a wide range of U, which is sum-

marized in the phase diagram in Fig. 1. The red dashed line

arXiv:2210.02662v1 [cond-mat.str-el] 6 Oct 2022

4.0 4.5 5.0 5.5 6.0

0.0

0.1

0.2

0.3

0.4

Metal

Band insulator

AF Mott insulator

FIG. 1. (Color online) Phase diagram of the Hubbard model on a

N=2L2honeycomb lattice at half ﬁlling. xrepresents the percentage

of the number of free sites with respect to the total number of lattice

sites, and Ulabels the on-site Coulomb repulsive interaction. Phase

boundaries are determined by the conductivity, density of states at

the Fermi energy and ﬁnite size scaling of the AFM spin structure

factor. There exists an intermediate phase −− band insulator −−

between the AFM Mott insulator and metal. These phases are labeled

by different colors in the ﬁgure: AFM Mott insulator (green), band

insulator (blue), metal (pink).

represents the phase boundary between the gapped insulator

and metal, which is determined by the conductivity and den-

sity of states. The black solid line indicates an AFM phase

transition conﬁrmed by the spin structure factor. Our results

have the possibility to be realized in optical lattice experi-

ments and were previously demonstrated in a one-dimensional

optical lattice to induce multiple phase transitions by introduc-

ing randomness into the interaction distribution via Feshbach

resonances[39,40].

II. MODEL AND METHOD

We consider a modiﬁed version of the Hubbard model with

site-dependent repulsion, described by the Hamiltonian:

H=−t∑

i∈A,j∈B,σ

(ˆc†

iσˆcjσ+H.c.)−µ∑

iσ

ˆniσ

+∑

Ui(ˆni↑−1

2)( ˆni↓−1

2).(1)

Here, ˆc†

iσ(ˆcj)indicates creation (annihilation) electron op-

erators in second-quantized formalism, and ˆniσ=ˆc†

iσˆciσis

the occupancy number operator. The ﬁrst term on the right-

hand side of Eq. (1) denotes in-plane hopping between near-

est neighbors, and in our paper, the hopping amplitude is set

as t=1, thus deﬁning the energy scale. The last term in-

cludes the chemical potential µ, and we set µ=0, a choice

that makes the studied system precisely half-ﬁlled and pro-

tects particle-hole symmetry.

We introduce dilution by allowing for random distribution

of the site-dependent Coulomb repulsion Uiin the second

term, such that the on-site interaction on a fraction x of the

sites is suppressed:

Ui=U1−x

This type of disorder is generated in a canonical ensemble,

i.e., we have a fraction Nx of sites with U=0 for a given con-

centration x, where N is the total number of sites, so that there

are no charge ﬂuctuations on the free sites. Here, we consider

a 2L2honeycomb lattice with a linear size of L=12. For di-

lution concentration x, where Nx is not an integer number, we

calculate a weighted average of its adjacent integers. Our re-

sults are obtained by averaging over 20 disorder realizations.

We probe the transport and magnetic properties of the half-

ﬁlled diluted honeycomb model by means of DQMC sim-

ulations. In this method, the Hamiltonian is mapped onto

free fermions moving in a ﬂuctuating space- and imaginary

time-dependent auxiliary ﬁeld by the Hubbard-Stratonovich

(HS) transform[41,42]. This HS ﬁeld is initialized randomly,

and a local ﬂip is attempted with the acceptance rate deter-

mined by the Metropolis algorithm. A QMC sweep is com-

pleted when the process of changing the auxiliary ﬁeld vari-

able traverses the entire space-time. In our simulations, 4,000

warm-up sweeps were used to equilibrate the system, and then

48,000 sweeps were conducted for measurements. The num-

ber of measurements was split into 10 bins, which provide

the basis of coarse-grain averages and errors estimated based

on standard deviations from the averages. The errors from

the Suzuki-Trotter decomposition are proportional to (∆τ)2,

where ∆τ=β/Mis the imaginary-time interval, so we set

∆τ=0.1 to guarantee that the systematic errors are smaller

than those associated with statistical sampling[43]. As with

many fermionic QMC methods, the DQMC method also suf-

fers from the minus-sign problem; however, the particle-hole

symmetry makes our system free of the sign problem so that

the simulation can be performed at low enough temperature to

converge to the ground state.

With the aim of exploring the phase transitions between

metal and insulating phases, we compute the T-dependent

direct-current conductivity:

σdc(T) = β2

πΛxx(q=0,τ=β/2).(2)

where β=1/Tis the inverse temperature and the mo-

mentum q- and imaginary time t-dependent current-current

correlation functions Λxx(q,τ)are expressed as Λxx(q,τ)

=hˆ

jx(q,τ)ˆ

jx(−q,0)i.ˆ

jx(q,τ)is the Fourier transform of

the τ-dependent current density operator in the x direction.

This approximation has been extensively employed to iden-

tify metal-insulator transitions for either disordered or clean

systems[44,45]. To establish the existence of the Mott insula-

tor, we deﬁne N(0), the density of states at the Fermi energy,

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AnintermediatephaseinducedbydilutioninacorrelatedDiracFermisystemLingyuTian,1,2JingyaoMeng,1,2andTianxingMa1,1DepartmentofPhysics,BeijingNormalUniversity,Beijing100875,China2BeijingComputationalScienceResearchCenter,Beijing100193,ChinaSubstitutingmagneticionswithnonmagneticionsisanewwaytostudydilut...

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An intermediate phase induced by dilution in a correlated Dirac Fermi system Lingyu Tian1 2Jingyao Meng1 2and Tianxing Ma1 1Department of Physics Beijing Normal University Beijing 100875 China

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